A Remarkable Connection

Natalie Paquette’s essay relates a remarkable and often ignored connection between physics and mathematics that has been established in recent years. Traditionally, this connection, the bridge, was considered to be a one-way street: physicists used results obtained by mathematicians in order to solve problems of interest to physics. They have been very successful in doing so. Group theory is a good example. Starting from the geometrical transformations in ordinary space, the subject grew into a rich chapter of abstract mathematics. Physicists found ready-made results that turned out to have unexpected applications in physics. Practically every one of the two hundred and thirty discrete subgroups of the group of isometries of three dimensional Euclidean space turned out to be the symmetry group of a crystal. An understanding of the fundamental interactions among elementary particles would have been impossible without abstract group theory. The quantum mechanical commutation relations motivated the study of deformations of groups and algebras which, in turn, found applications in physical systems.

With the advent of string theory, as Paquette points out, the bridge initiated two-way communication. Physicists applied concepts and methods developed by physicists in order to answer questions directly related to physics, and obtained results of interest to pure mathematics. These contributions were acknowledged by the community of mathematicians, who awarded the Fields medal to Edward Witten, a prominent theoretical physicist. It was the first time that this honor, the highest scientific award for mathematicians, was given to a scientist from the other side of the bridge.

Paquette’s essay is a beautifully written account by a researcher that has directly contributed to this connection. But, I would not describe it as easy to read. The language is quite technical and the author makes no concessions regarding precision and rigor. A layman unfamiliar with superstring theory or algebraic geometry will have difficulty in following the arguments. Nonetheless, they would be wrong to give up. If the layman is able to bypass the technical details, the central idea is clearly explained. The reward for their persistence will be a new vision that goes beyond the artificial boundaries that have been erected between fields of knowledge in our current era of overspecialization. The lay reader can then share in the enthusiasm and delight that the author feels in discovering this profound unity of human thought.

The clear distinction between mathematics and theoretical physics is, in fact, a rather recent development in the history of science. Both disciplines can claim Isaac Newton: as a mathematician for differential calculus, and as a theoretical physicist for Philosophiæ Naturalis Principia Mathematica. William Hamilton, working at the middle of the nineteenth century, would not be out of place at either end of the bridge. In the early-twentieth century, Emmy Noether was an untenured professor in the mathematics department at the University of Göttingen, but the theorem that bears her name is most often found in physics textbooks. From an epistemological point of view, this interconnection is not surprising. All the fields of science originated from the efforts made by the early scientists to put some order in the surrounding world and establish relations between observations and measurements. The needs of everyday life were often the first motivation. It seems that the earliest results in geometry were obtained by the clerks of ancient Egypt, who were tasked with an annual evaluation of the land submerged by the Nile when it flooded. The reason was taxation. The Pharaoh wanted to tax only the land that was effectively cultivated. The exponential function was introduced by Sumerian bankers in ancient Mesopotamia. They were the first to use compound interest. In both cases, the motivations were very serious: they had to do with money. Of course, after a certain point, each discipline acquired its own momentum, developed a proper methodology and formulated its problems internally, without a need for continuous outside influences. When the Greeks picked up geometry they turned it into a field of pure mathematics, the first axiomatic system in the history of science based on the laws of logic. In this sense, the connection between mathematics and theoretical physics can be understood from the fact that they are both manifestations of the logical character of the human mind.

This real connection is often forgotten, especially in our educational system. When young mathematicians learn some physics, they consider it only as a demonstration of the ability of mathematics to make contact with the real world. Similarly, physicists perceive mathematics as merely a tool to obtain a precise description of physical phenomena. As Paquette points out, this is a one-way bridge. What is rarely emphasized is that, through their mutual interactions, these two disciplines have enriched themselves and shaped our understanding of the fundamental laws of nature. The examples presented are eloquent.

Paquette concludes her essay with a question: is it possible that the recent revival of this deep connection is not merely an accident, but instead signals a real change in the way physicists and mathematicians look at each other?

I, for one, hope that she will be proven right.

John Iliopoulos

John Iliopoulos is a CNRS Director of Research Emeritus at the Ecole Normale Supérieure in Paris, where for many years, he was the head of the Theoretical Physics Department.